On some multiplicity and mixed multiplicity formulas (Forum Math. 26(2014), 413-442)

This paper gives the additivity and reduction formulas for mixed multiplicities of multi-graded modules $M$ and mixed multiplicities of arbitrary ideals, and establishes the recursion formulas for the sum of all the mixed multiplicities of $M\mathrm{.}$ As an application of these formulas we get the recursion formulas for the multiplicity of multi-graded Rees modules.